Statistics · Class 10 Mathematics: quick revision, 51 bites

Why grouping needs an assumption

The values are gone

A class tells you how many observations fell in it, not what they were.

Takeaway. Grouping trades detail for readability.

The class mark

Stand them at the middle

Takeaway. Every observation in a class is treated as sitting here.

The direct method

Method 1

Takeaway. Multiply, add, divide by the total frequency.

Our thirty students

The direct method at work

Takeaway. Sum of f·x over sum of f.

The assumed mean

Method 2

Takeaway. Measure everything FROM a.

The algebra behind it

Why a can be anything

Takeaway. The a subtracted comes straight back.

Step deviation

Method 3

Takeaway. Needs classes of EQUAL width h.

u values −2 to 2

Step deviation at work

Takeaway. Small enough to do in your head.

What u means

Just a change of units

$u_i$ measures each class mark in CLASS-WIDTHS away from $a$.

Takeaway. That h is not optional.

Forgetting the h

The commonest slip

Omitting $h$ in the step-deviation formula makes the answer far too small.

Takeaway. Check the size of your answer against the class marks.

The denominator

Divide by what?

The divisor is $\sum f_i$, the TOTAL FREQUENCY — never the number of classes.

Takeaway. 900 over 30, not 900 over 5.

Exactly, not roughly

All three agree

Direct, assumed mean and step deviation give the SAME answer.

Takeaway. Use whichever keeps the arithmetic smallest.

Choosing a method

Which to use

Takeaway. Unless the question names one, any is acceptable.

Where the marks are

Build the table

Columns for $x_i$, $f_i$ and $f_i x_i$ — or $u_i$ and $f_i u_i$ — with a TOTAL row.

Takeaway. The table is the answer; substituting is one line.

d is x minus a

Mind the sign

Taking $d_i = a - x_i$ reverses every sign and moves the mean the wrong way.

Takeaway. Always the class mark minus a.

What the answer really is

An estimate

The individual values were discarded, so the result approximates the true mean.

Takeaway. Saying so costs nothing and is honest.

A word from the sea

Where 'average' came from

It traces back to an Arabic term for damage to cargo, shared out among merchants.

Takeaway. The idea of sharing equally is the old one.

What a mode is

The most common value

The observation that occurs most often.

Takeaway. Counting, not arithmetic.

The modal class

The tallest bar

The class with the largest frequency.

Takeaway. Easy to spot in a histogram.

Modal class against mode

The class is not the answer

The modal class is an interval; the MODE is a single value inside it.

Takeaway. Name the class, then interpolate.

Mode of grouped data

The one formula

Takeaway. The whole topic is reading its five letters correctly.

Five numbers

What the letters mean

Takeaway. Write all five down before substituting.

What l is

Not the class mark

$l$ is the LOWER BOUNDARY of the modal class, not its midpoint.

Takeaway. For 30-40 it is 30, not 35.

Mixing up f0 and f2

Before and after

Swapping them pushes the mode to the wrong side of the class.

Takeaway. Point at the row above and the row below.

Modal class 30-40

A worked mode

Takeaway. Low in the class, because the class below is nearly as tall.

The fraction is between 0 and 1

Why it always lands inside

With $A = f_1 - f_0$ and $B = f_1 - f_2$, both positive, the fraction is $\frac{A}{A+B}$.

Takeaway. So the mode is always within the modal class.

Is it inside?

A five-second check

The mode must lie between the two boundaries of the modal class.

Takeaway. Run this check before writing the answer down.

Towards the shorter neighbour

Which way it leans

A tall class BEFORE keeps the mode low; a tall class AFTER pushes it high.

Takeaway. The formula encodes the crowding.

Missing neighbours

At either end

Modal class first: $f_0 = 0$.

Takeaway. There is no row above the first or below the last.

12, 8, 5

A first-class mode

Takeaway. f0 is zero because nothing comes before.

The denominator

Two f-ones

It is $2f_1 - f_0 - f_2$, with the modal frequency counted TWICE.

Takeaway. Not $f_1 - f_0 - f_2$.

Bimodal data

Two tallest classes

A tie for the largest frequency leaves no single modal class.

Takeaway. Check the frequencies again.

Discontinuous classes

Make them meet first

0-9, 10-19 must become -0.5-9.5, 9.5-19.5 before $l$ and $h$ are read off.

Takeaway. Adjust, then substitute.

Where modes matter

Which size to stock

A shoe shop cares about the most common size, not the average one — half sizes nobody wears are no use.

Takeaway. Sometimes the commonest value beats the typical one.

What a median is

The middle one

Arrange the data in order; the median is the observation in the middle.

Takeaway. Position, not size.

Why medians are reported

Extremes cannot shift it

Multiplying the largest value by ten moves the mean but leaves the median exactly where it was.

Takeaway. Which is why incomes and house prices use it.

Cumulative frequency

The running total

The total frequency up to and including each class.

Takeaway. Build this column first, every time.

Finding the median class

Halfway down

Compute $\frac{n}{2}$, then run down the cumulative column.

Takeaway. For grouped data it is n/2, not (n+1)/2.

Median of grouped data

The one formula

Takeaway. Four numbers, read from the table.

What cf means

From the row above

$cf$ is the cumulative frequency of the class BEFORE the median class.

Takeaway. The commonest error in the topic.

What f means

Its own, not cumulative

$f$ is the median class's OWN frequency — how many observations that class holds.

Takeaway. Not the running total.

Median class 30-40

A worked median

Takeaway. Just inside, because 14 of the 30 were already counted.

What the formula is saying

Read it as an instruction

You need observation number $\frac{n}{2}$; the earlier classes gave $cf$ of them.

Takeaway. Walk that fraction of the way in from l.

Is it inside?

A five-second check

By definition $cf < \frac{n}{2} \le cf + f$, so the fraction is between 0 and 1.

Takeaway. The median must lie inside the median class.

When cf is zero

At the start

If the median class is the FIRST class, there is no class before it and $cf = 0$.

Takeaway. Nothing was counted before it.

The empirical relation

Tying the three together

Takeaway. An approximation, found by observation, not proof.

Our table

How close is it?

Mean 30, median 31, so the relation estimates the mode as $3(31) - 2(30) = 33$.

Takeaway. Close, but not equal — and that is the point.

Getting the other two

Turned round

Takeaway. Two given, one wanted.

Symmetric distributions

When it is exact

If mean, median and mode coincide, the relation reads $3M - 2M = M$ ✓

Takeaway. Symmetry is what makes it exact.

Choosing one to quote

Which average?

Takeaway. They answer different questions.

Ogives are out

Not a graph

Cumulative frequency is used here only as a COLUMN in the table.

Takeaway. Build the column, not the curve.